Tricker Arcs & Anthelic Reflection Dynamics: How Horizontally Oriented Columnar Crystals and Basal-Prism Internal Paths Forge Crossed Antisolar Loops

1. Opening Scene

The thermometer bolted to the timber of the weather shelter reads $-28^\circ\text{C}$, but the cold registers less as a number than as an immediate physical reality. In this sub-zero plateau, the air has been scrubbed clean of all humidity save for a suspended suspension of microscopic ice crystals—an ethereal veil known to polar meteorologists as diamond dust. When you inhale, the desiccated air bites against the membranes of your throat with a dry, metallic sting. The wind is completely dead; the local atmosphere has settled into an inversion layer so quiet that the faint clinking of ice rime falling from the guy-wires sounds like shattered glass.

Looking toward the south, the low winter sun hangs barely twelve degrees above the serrated horizon, burning with an intense, pale glare. It is flanked by the familiar, blazing apparitions of the $22^\circ$ parhelia (sundogs) and a towering light pillar piercing straight toward the zenith. But the true optical marvel of this frozen morning does not lie in the solar sky.

Turn your back entirely to the sun. Turn one hundred and eighty degrees until your shadow stretches out before you across the pristine snow crust. Directly ahead, centered on your shadow's head at the anti-solar point, the sky should be empty blue. Instead, floating in the crystal haze, lies a bright, diffuse knot of pure white light: the anthelion.

Shooting out from this ghost sun are two sweeping, razor-sharp ribbons of luminescence. They cross one another at the anthelion, forming an immense celestial figure-eight that curves upward, loops overhead across the zenith, and descends back toward the flanks of the parhelic circle. These are Tricker arcs. They do not shimmer with the split-spectrum prismatic fire of a rainbow; they glow with a stark, silver intensity—the hallmark of light that has undergone a journey of total internal reflection inside billions of aerodynamically suspended hexagonal crystals.

2. What's Actually Happening — Plain English First

To understand how sunlight can bounce backward across the sky and tie itself into symmetrical knots behind you, imagine the sky filled not with formless vapor, but with millions of microscopic, perfectly carved glass pencils.

Each of these ice crystals is a six-sided hexagonal prism terminated by flat, perpendicular end-caps. In turbulent, stormy air, these tiny crystals tumble erratically like confetti, smearing light into diffuse, featureless halos. But in the laminar stillness of a freezing morning, aerodynamics takes over. Think of a wooden pencil dropped into a swimming pool: rather than falling point-first like an arrow, it settles through the water horizontally, presenting its broad side to the fluid beneath it to maximize resistance.

In identical fashion, as these microscopic ice columns settle gently through the dense, frigid air, aerodynamic drag forces their long axes to lie parallel to the earth's surface. They are free to point their tips north, south, east, or west, but they refuse to tilt vertically. Meteorologists call this state singly oriented horizontal column orientation.

Now, track a single beam of light entering one of these floating crystals:

1. The Entry: The sunbeam strikes one of the sloping side faces (a prism facet) of a horizontal crystal. As it crosses from air into ice, the beam bends (refracts) downward and inward.
2. The Internal Mirror: Instead of passing straight out the other side, the angled ray travels down the length of the crystal tube and strikes the flat, polished end-cap (the basal facet). Because the light hits this interior boundary at a shallow, grazing angle, the glass-like ice surface acts as a mirror through a phenomenon called total internal reflection. Not a single photon escapes out the end; the entire beam is bounced backward along the crystal's interior.
3. The Exit: The reflected beam hits an opposite side face and refracts out into the open sky.

Because the crystal's end-face acted as an internal retroreflector, the light emerges traveling in the direction opposite the sun. As millions of these horizontal crystals drift across the sky at every compass orientation, their collective reflections weave together into sweeping arcs that intersect at the anti-solar horizon—the anthelic point.

3. The Science (for those who want to go deeper)

The physics of anthelic halos represents one of the most elegant intersections of fluid dynamics, crystallography, and 3D vector optics in terrestrial meteorology. While forward-scattering phenomena such as the $22^\circ$ halo and parhelia are cataloged in standard references like the WMO International Cloud Atlas, anthelic arcs belong to a class of complex ray paths first systematically classified by British atmospheric physicist R. A. R. Tricker in 1970.

The Hierarchy of Anthelic Halo Phenomena

When columnar crystals align horizontally, they generate several distinct families of anthelic and sub-anthelic arcs, categorized by their internal ray trajectories:

As documented by the National Oceanic and Atmospheric Administration (NOAA) Arctic research teams, Tricker arcs require exceptionally high crystal perfection. Surface pitting, hollow ends, or rounding of crystal edges destroys the total internal reflection condition, extinguishing the arcs.

Aerodynamic Stability in Laminar Boundary Layers

Why do columnar ice crystals settle with their long axis horizontal? The answer lies in low-Reynolds-number hydrodynamics. For a hexagonal ice cylinder of length $L \approx 100\,\mu\text{m}$ and diameter $d \approx 30\,\mu\text{m}$ falling through air at $T = -25^\circ\text{C}$ ($\rho_{\text{air}} \approx 1.43\,\text{kg/m}^3$, dynamic viscosity $\mu \approx 1.58 \times 10^{-5}\,\text{Pa}\cdot\text{s}$), the terminal settling velocity $U$ is on the order of $0.05\,\text{m/s}$.

The Reynolds number governing this settling flow is:

$$Re = \frac{\rho_{\text{air}} U d}{\mu} \approx \frac{1.43 \times 0.05 \times (30 \times 10^{-6})}{1.58 \times 10^{-5}} \approx 0.136$$

In this creeping-to-transitional flow regime ($0.1 < Re < 5$), inertial effects in the thin boundary layer around the crystal create an asymmetric pressure distribution if the cylinder tilts. Any angular deflection $\theta_{\text{tilt}}$ away from the horizontal produces a restoring aerodynamic torque $\mathbf{\tau}_{\text{aero}}$:

$$\mathbf{\tau}{\text{aero}} \propto -\rho{\text{air}} U^2 L d^2 \sin(2\theta_{\text{tilt}})$$

This torque forces the principal hexagonal optic axis ($c$-axis) to lie strictly within the horizontal plane ($x$-$y$ plane), leaving the rotational angle $\alpha$ around the vertical $z$-axis uniformly distributed over $[0, 2\pi)$.

3D Vector Refraction and Total Internal Reflection Dynamics

To trace a ray through a horizontal pencil crystal, we define an orthonormal frame where the unit vector $\mathbf{\hat{c}}$ along the crystal's long axis lies in the horizontal plane:

$$\mathbf{\hat{c}} = (\cos\alpha, \sin\alpha, 0)$$

Let the incoming solar ray vector be $\mathbf{\hat{s}}$, defined by solar elevation $\Sigma$ and solar azimuth (taken as $0^\circ$ without loss of generality):

$$\mathbf{\hat{s}} = (-\cos\Sigma, 0, -\sin\Sigma)$$

Vector Snell's Law in 3D Space

When a unit ray vector $\mathbf{\hat{v}}i$ strikes a crystal facet with inward surface unit normal $\mathbf{\hat{n}}$, the refracted unit vector $\mathbf{\hat{v}}_r$ inside the ice (with refractive index $n{\text{ice}} \approx 1.309$ at $\lambda = 589\,\text{nm}$) is given by:

$$\mathbf{\hat{v}}r = \frac{1}{n{\text{ice}}} \mathbf{\hat{v}}i + \left( \sqrt{1 - \frac{1}{n{\text{ice}}^2}\left(1 - (\mathbf{\hat{v}}i \cdot \mathbf{\hat{n}})^2\right)} - \frac{1}{n{\text{ice}}}(\mathbf{\hat{v}}_i \cdot \mathbf{\hat{n}}) \right) \mathbf{\hat{n}}$$

The Total Internal Reflection (TIR) Condition at the Basal Facet

Inside the crystal, the ray travels until it encounters the basal end-cap facet, whose outward surface normal is parallel to $\mathbf{\hat{c}}$ (hence the internal normal is $-\mathbf{\hat{c}}$).

For total internal reflection to occur at this basal boundary, the angle of incidence inside the ice, $\theta_{\text{int}} = \arccos(|\mathbf{\hat{v}}_r \cdot \mathbf{\hat{c}}|)$, must exceed the critical angle $\theta_c$:

$$\sin\theta_c = \frac{n_{\text{air}}}{n_{\text{ice}}} = \frac{1.000}{1.309} \implies \theta_c = \arcsin\left(\frac{1}{1.309}\right) \approx 49.81^\circ$$

Because the crystal's longitudinal axis is horizontal, the axial component of the ray vector inside the ice preserves the projection:

$$\mathbf{\hat{v}}r \cdot \mathbf{\hat{c}} = \frac{1}{n{\text{ice}}} (\mathbf{\hat{s}} \cdot \mathbf{\hat{c}})$$

The internal angle relative to the basal facet normal is $\phi_{\text{basal}} = \frac{\pi}{2} - \theta_{\text{int}}$. Reflection off this basal wall specularly inverts the axial component of the ray:

$$\mathbf{\hat{v}}_r' = \mathbf{\hat{v}}_r - 2(\mathbf{\hat{v}}_r \cdot \mathbf{\hat{c}})\mathbf{\hat{c}}$$

The ray reverses its longitudinal course, travels toward the opposite end of the crystal, strikes an opposing prism facet (oriented at $60^\circ$ or $120^\circ$ to the entry facet), and refracts out into the atmosphere.

Worked Numerical Example: The Anthelic Crossing Condition

Let us trace a concrete ray bundle to show why these rays converge at the anthelic point.

Assume: * Solar Elevation: $\Sigma = 15.0^\circ$ ($\mathbf{\hat{s}} = (-\cos 15^\circ, 0, -\sin 15^\circ) = (-0.9659, 0, -0.2588)$) * Refractive Index of Ice: $n = 1.309$ * Consider a crystal whose long axis is oriented at an azimuth $\alpha = 45.0^\circ$ to the sun: $$\mathbf{\hat{c}} = (\cos 45^\circ, \sin 45^\circ, 0) = (0.7071, 0.7071, 0)$$

Step 1: Axial Projection Compute the scalar product of the solar ray with the crystal axis: $$\mathbf{\hat{s}} \cdot \mathbf{\hat{c}} = (-0.9659)(0.7071) + (0)(0.7071) + (-0.2588)(0) = -0.6830$$

Inside the ice, the axial component along $\mathbf{\hat{c}}$ is: $$(\mathbf{\hat{v}}_r \cdot \mathbf{\hat{c}}) = \frac{-0.6830}{1.309} = -0.5218$$

Step 2: Check Basal Total Internal Reflection The angle of incidence onto the basal facet inside the crystal is: $$\theta_{\text{basal}} = \arccos(|\mathbf{\hat{v}}_r \cdot \mathbf{\hat{c}}|) = \arccos(0.5218) \approx 58.54^\circ$$

Since $\theta_{\text{basal}} = 58.54^\circ > \theta_c = 49.81^\circ$, total internal reflection is guaranteed. The reflection coefficient is identically $1.000$ (100% reflectance), preserving the brilliance of the ray.

Step 3: Basal Reflection Transformation Upon reflection, the axial vector component changes sign: $$(\mathbf{\hat{v}}_r' \cdot \mathbf{\hat{c}}) = +0.5218$$

Step 4: Emergent Ray Alignment After the ray exits through the complementary side prism facet, the cross-sectional refraction cancels the transverse deviation while reversing the longitudinal direction. The emergent ray $\mathbf{\hat{e}}$ has the elevation angle: $$\sin \Sigma_{\text{exit}} = -\sin \Sigma_{\text{sun}} = -\sin(15^\circ) = -0.2588 \implies \Sigma_{\text{exit}} = -15.0^\circ$$

Relative to an observer looking toward the anti-solar horizon ($180^\circ$ from the sun), this ray appears at an altitude of $+15.0^\circ$ above the horizon—precisely intersecting the anthelion.

Morphology Across Varying Solar Elevations

The visual morphology of Tricker arcs changes dramatically as the sun climbs in the sky:

• Low Solar Elevation ($\Sigma < 10^\circ$): The arcs appear as two towering, nearly vertical luminous pillars intersecting at the anthelion, sweeping up to form broad, sweeping loops that cross the zenith and extend deep into the solar half of the sky.
• Moderate Solar Elevation ($10^\circ \le \Sigma \le 25^\circ$): The loops widen into a dramatic lateral figure-eight. The anthelic crossing point remains intensely sharp, and the outer wings of the arcs tangentially brush the parhelic circle.
• High Solar Elevation ($\Sigma > 35^\circ$): Total internal reflection fails for a growing proportion of crystal azimuths because the internal angle drops below $\theta_c \approx 49.81^\circ$. The Tricker arcs fade into faint, ghostly loops detached from the anthelion before vanishing entirely when $\Sigma \ge 42^\circ$.

4. Practical Outdoor Guidance

Observing Tricker arcs in the field is one of the most rewarding challenges in atmospheric optics. Because they require horizontal column crystals, they rarely appear in warm-season cirrus clouds; instead, they are almost exclusively found in low-level diamond dust plumes in polar regions, alpine basins, or during extreme continental cold waves.

What to Look For in the Sky

1. Locate the Parhelic Circle: Find the horizontal white band of light that passes through the sun and runs parallel to the horizon at the same altitude as the sun.
2. Trace to the Anthelion: Turn $180^\circ$ and follow the parhelic circle to the anti-solar point. Look for a bright, uncolored white spot on the circle directly opposite the sun.
3. Inspect the Anthelic Intersection: Search for an oblique "X" crossing directly through the anthelion. The arms of this cross that curve upward and loop over the zenith are Tricker arcs. (Wegener arcs also pass through this region, but they do not form the distinct closed lateral loops characteristic of Tricker paths; see the Met Office Guide to Atmospheric Optics).

Critical Meteorological Indicators

• Surface Temperature: $T \le -15^\circ\text{C}$ ($5^\circ\text{F}$), with optimum crystal growth occurring between $-20^\circ\text{C}$ and $-35^\circ\text{C}$. At these temperatures, atmospheric moisture deposits predominantly as pristine columnar prisms rather than plates or stellar dendrites.
• Barometric Profile: Look for a strong, stable continental anticyclone (barometer reading $> 1025\,\text{hPa}$). The resulting surface radiation inversion creates the dead-calm, laminar conditions necessary for aerodynamic crystal alignment.
• Surface Winds: Wind speeds must be under $1.5\,\text{m/s}$ (3 knots). Turbulence from mechanical wind shear destroys the horizontal alignment of the column crystals, scattering the coherent arcs into a blurry, indistinct fog.

Field Observation Techniques

• Polarized Eyewear: Because Tricker arcs rely on total internal reflection and grazing-angle refractions, their light is strongly linearly polarized perpendicular to the arc trajectories. Rotating a polarizing filter while viewing the anthelic sky will make the arcs alternate between near-total extinction and vivid contrast against the background ice haze.
• Solar Occlusion: Always shield your eyes and camera lens from the direct sun using a physical obstacle (such as the corner of a building, a tree trunk, or an outstretched mitten). This prevents glare from washing out the subtle, diffuse anthelic loops in the opposite quadrant.

5. Today's Meteorological Rule of Thumb

When diamond dust falls through dead-calm air at minus twenty, turn your back on the sun: the rarest wonders of atmospheric reflection are waiting in your shadow.

Authoritative References and Further Reading

• World Meteorological Organization (WMO) International Cloud Atlas — Classification of Ice Crystal Hydrometeors and Optical Phenomena.
• National Oceanic and Atmospheric Administration (NOAA) — Arctic Meteorology and Polar Atmospheric Radiation Studies.
• Met Office (UK) — Optical Phenomena in the Atmosphere.
• Halo Optical Phenomena (Wikipedia Overview) — Structural hierarchy of refraction and reflection arcs.
• Tricker Arc Mechanics (Wikipedia) — Ray path diagrams and historical classification.
• Total Internal Reflection in Hexagonal Media — Boundary physics in atmospheric crystallography.